Semiconductor qubits for quantum computation

1. Semiconductor qubits for quantum computation Matthias Fehr TU Munich JASS 2005 St.Petersburg/Russia Is it possible to realize a quantum computer with semiconductor technology?

2. Contents • History of quantum computation • Basics of quantum computation • Qubits • Quantum gates • Quantum algorithms • Requirements for realizing a quantum computer • Proposals for semiconductor implementation • Kane concept: Si:31P • Si-Ge heterostructure • Quantum Dots (2D-electron gas, self-assembly) • NV- center in diamond

3. 1936 Alan Turing 1982 Feynman 1985 Deutsch Church-Turing thesis: There is a „Universal Turing machine“, that can efficiently simulate any other algorithm on any physical device Computer based on quantum mechanics might avoid problems in simulating quantum mech. systems Search for a computational device to simulate an arbitrary physical system quantum mechanics -> quantum computer Efficient solution of algorithms on a quantum computer with no efficient solution on a Turing machine? The history of quantum computation

4. 1994 Peter Shor 1995 Lov Grover In the 1990s 1995 Schumacher 1996 Calderbank, Shor, Steane Efficient quantum algorithms prime factorization discrete logarithm problem ->more power Efficient quantum search algorithm Efficient simulation of quantum mechanical systems “Quantum bit” or “qubit” as physical resource Quantum error correction codes - protecting quantum states against noise The history of quantum computation

5. Classical bit: 0 or 1 2 possible values Qubit: are complex -> infinite possible values -> continuum of states The basics of quantum computation Qubit measurement: result 0 with probability result 1 with probability Wave function collapses during measurement, qubit will remain in the measured state

6. Qubits Bloch sphere: We can rewrite our state with phase factors Qubit realizations: 2 level systems 1) ground- and excited states of electron orbits 2) photon polarizations 3) alignment of nuclear spin in magnetic field 4) electron spin … Bloch sphere [from Nielson&Chuang]

7. Single qubit gates • Qubits are a possibility to store information quantum mechanically • Now we need operations to perform calculations with qubits • -> quantum gates: • NOT gate: classical NOT gate: 0 -> 1; 1 -> 0 quantum NOT gate: • Linear mapping -> matrix operation Equal to the Pauli spin-matrix

8. Single qubit gates • Every single qubit operation can be written as a matrix U • Due to the normalization condition every gate operation U has to be unitary • -> Every unitary matrix specifies a valid quantum gate • Only 1 classical gate on 1 bit, but quantum gates on 1 qubit. • Z-Gateleaves unchanged, and flips the sign of • Hadamard gate = “square root of NOT”

9. Hadamard gate • Bloch sphere: - Rotation about the y-axis by 90° - Reflection through the x-y-plane • Creating a superposition Bloch sphere [from Nielson&Chuang]

10. Decomposing single qubit operations • An arbitrary unitary matrix can be decomposed as a product of rotations • 1st and 3rd matrix: rotations about the z-axis • 2nd matrix: normal rotation • Arbitrary single qubit operations with a finite set of quantum gates • Universal gates

11. Multiple qubits For quantum computation multiple qubits are needed! 2 qubit system: computational bases stats: superposition: Measuring a subset of the qubits: Measurement of the 1st qubit gives 0 with probability leaving the state 1st qubit 2nd qubit

12. Entanglement • Bell state or EPR pair: prepare a state: • Measuring the 1st qubit gives 0 with prop. 50% leaving 1 with prop. 50% leaving • The measurement of the 2nd qubits always gives the same result as the first qubit! • The measurement outcomes are correlated! • Non-locality of quantum mechanics • Entanglement means that state can not be written as a product state

13. Multiple qubit gates, CNOT • Classical: AND, OR, XOR, NAND, NOR -> NAND is universal • Quantum gates: NOT, CNOT • CNOT gate: - controlled NOT gate = classical XOR - If the control qubit is set to 0, target qubit is the same - If the control qubit is set to 1, target qubit is flipped • CNOT is universal for quantum computation • Any multiple qubit logic gate may be composed from CNOT and single qubit gates • Unitary operations are reversible (unitary matrices are invertible, unitary -> too ) • Quantum gate are always reversible, classical gates are not reversible

14. Qubit copying? • classical: CNOT copies bits • Quantum mech.: impossible • We try to copy an unknown state • Target qubit: • Full state: • Application of CNOT gate: • We have successful copied , but only in the case • General state • No-cloning theorem: major difference between quantum and classical information

15. Quantum parallelism • Evaluation of a function: • Unitary map: black box • Resulting state: • Information on f(0) and f(1) with a single operation • Not immediately useful, because during measurement the superposition will collapse Quantum gate [from Nielson&Chuang (2)]

16. Deutsch algorithm • Input state: • Application of :

17. Deutsch algorithm • global property determined with one evaluation of f(x) • classically: 2 evaluations needed • Faster than any classical device • Classically 2 alternatives exclude one another • In quantum mech.: interference

18. Quantum algorithms

19. The Five Commandments of DiVincenzo • A physical system containing qubits is needed • The ability to initialize the qubit state • Long decoherence times, longer than the gate operation time • Decoherence time: 104-105 x “clock time” • Then error-correction can be successful • A universal set of quantum gates (CNOT) • Qubit read-out measurement

20. Systems have to be almost completely isolated from their environment The coherent quantum state has to be preserved Completely preventing decoherence is impossible Due to the discovery of quantum error-correcting codes, slight decoherence is tolerable Decoherence times have to be very long -> implementation realizable Performing operations on several qubits in parallel 2- Level system as qubit: Spin ½ particles Nuclear spins Read-out: Measuring the single spin states Bulk spin resonance Realization of a quantum computer

21. Si:31P, Kane concept from 1998 • Logical operations on nuclear spins of 31P(I=1/2) donors in a Si host(I=0) • Weakly bound 31P valence electron at T=100mK • Spin degeneracy is broken by B-field • Electrons will only occupy the lowest energy state when • Spin polarization by a strong B-field and low temperature • Long 31P spin relaxation time , due to low T

22. Single spin rotations • Hyperfine interaction at the nucleus • : frequency separation of the nuclear levels • A-gate voltage pulls the electron wave function envelope away from the donors • Precession frequency of nuclear spins is controllable • 2nd magnetic field Bac in resonance to the changed precession frequency • Selectively addressing qubits • Arbitrary spin rotations on each nuclear spin

23. Qubit coupling • J-gates influence the neighboring electrons -> qubit coupling • Strength of exchange coupling depends on the overlap of the wave function • Donor separation: 100-200 • Electrons mediate nuclear spin interactions, and facilitate measurement of nuclear spins

24. Qubit measurement • J < µBB/2: qubit operation • J > µBB/2: qubit measurement • Orientation of nuclear spin 1 alone determines if the system evolves into singlet or triplet state • Both electrons bound to same donor (D- state, singlet) • Charge motion between donors • Single-electron capacitance measurement • Particles are indistinguishable

25. Materials free of spin( isotopes) Ordered 1D or 2D-donor array Single atom doping methodes Grow high-quality Si layers on array surface 100-A-scale gate devices Every transistor is individual -> large scale calibration A-gate voltage increases the electron-tunneling probability Problems with low temperature environment Dissipation through gate biasing Eddy currents by Bac Spins not fully polarized Many problems

26. SRT with Si-Ge heterostructures • Spin resonance transistors, at a size of 2000 A • Larger Bohr radius (larger ) • Done by electron beam lithography • Electron spin as qubit • Isotropic purity not critical • No needed spin transfer between nucleus and electrons • Different g-factors Si: g=1.998 / Ge: g=1.563 • Spin Zeeman energy changes • Gate bias pulls wave function away from donor

27. Confinement and spin rotations • Confinement through B-layer • RF-field in resonance with SRT -> arbitrary spin phase change

28. 2-qubit interaction • No J-gate needed • Both wave functions are pulled near the B-layer • Coulomb potential weakens • Larger Bohr radius • Overlap can be tuned • CNOT gate

29. Detection of spin resonance • FET channel: n-Si0.4Ge0.6 ground plane counter-electrode • Qubit between FET channel and gate electrode • Channel current is sensitive to donor charge states: • ionized / neutral / doubly occupied (D- state) • D- state (D+ state) on neighbor transistors, change in channel current -> Singlet state • Channel current constant -> triplet state

30. Electro-statically defined QD • GaAs/AlGaAs heterostructure -> 2DEG • address qubits with • high-g layer • gradient B-field • Qubit coupling by lowering the tunnel barrier

31. Single spin read-out in QD • Spin-to-charge conversion of electron confined in QD (circle) • Magnetic field to split states • GaAs/AlGaAs heterostructure -> 2DEG • Dots defined by gates M, R, T • Potential minimum at the center • Electron will leave when spin- • Electron will stay when spin- • QPC as charge detector • Electron tunneling between reservoir and dot • Changes in QQPC detected by measuring IQPC

32. Two-level pulse on P-gate

33. Self-assembled QD-molecule • Coupled InAs quantum dots • quantum molecule • Vertical electric field localizes carriers • Upper dot = index 0 • Lower dot = index 1 • Optical created exciton • Electric field off -> tunneling -> entangled state

34. Self-assembled Quantum Dots array • Single QD layer • Optical resonant excitation of e-h pairs • Electric field forces the holes into the GaAs buffer • Single electrons in the QD ground state (remains for hours, at low T) • Vread: holes drift back and recombine • Large B-field: Zeeman splitting of exciton levels

35. Self-assembled Quantum Dots array • Circularly polarized photons • Mixed states • Zeeman splitting yields either • Optical selection of pure spin states

36. NV- center in diamond • Nitrogen Vacancy center: defect in diamond, N-impurity • 3A -> 3E transition: spin conserving • 3E -> 1A transition: spin flip • Spin polarization of the ground state • Axial symmetry -> ground state splitting at zero field • B-field for Zeeman splitting of triplet ground state • Low temperature spectroscopy

37. NV- center in diamond • Fluorescence excitation with laser • Ground state energy splitting greater than transition line with • Excitation line marks spin configuration of defect center • On resonant excitation: • Excitation-emission cycles 3A -> 3E • bright intervals, bursts • Crossing to 1A singlet small • No resonance • Dark intervals in fluorescence

38. Thank you very much! • Kane, B.E. A silicon-based nuclear spin quantum computer, Nature 393, 133-137 (1998) • Nielson & Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000) • Loss, D. et al. Spintronics and Quantum Dots, Fortschr. Phys. 48, 965-986, (2000) • Knouwenhoven, L.P. Single-shot read-out of an individual electron spin in a quantum dot, Nature 430, 431-435 (2004) • Forchel, A. et al., Coupling and Entangling of Quantum States in Quantum Dot Molecules, Science 291, 451-453 (2001) • Finley, J. J. et al., Optically programmable electron spin memory using semiconductor quantum dots, Nature 432, 81-84 (2004) • Wrachtrup, J. et al., Single spin states in a defect center resolved by optical spectroscopy, Appl. Phys. Lett. 81 (2002) • Doering, P. J. et al. Single-Qubit Operations with the Nitrogen-Vacancy Center in Diamond, phys. stat. sol. (b) 233, No. 3, 416-426 (2002) • DiVincenzo, D. et al., Electron Spin Resonance Transistors for Quantum Computing in Silicon-Germanium Heterostructures, arXiv:quant-ph/9905096 (1999) • DiVincenzo, D. P., The Physical Implementation of Quantum Computation, Fortschr. Phys. 48 (2000) 9-11, 771-783